With the analysis of the finite length coded FTN system which considers full ISI taps, we can now investigate the error performance for the ML estimate when a finite number of ISI taps \(L_E < L\)  is considered by the detector.
As in the previous section, we firstly consider the case of uncoded finite-taps FTN signaling. Following [1]} and Definition 2,
we intend to find the error probability \(P_{\mathbf {e}}^{^{\prime }}\)  under the condition that a finite number of ISI responses is considered at the receiver, which is equivalent of finding the probability of \(P(\text{Re}\big {\lbrace } \mathbf {e}^\text{H}{\eta } + \mathbf {e}^\text{H}(\mathbf {G}-\mathbf {F})\mathbf {x}\big {\rbrace  > E_bd^2_{\text{ope}}(\mathbf {e})})\) .
It can be seen that \(\mathbf {e}^\text{H}(\mathbf {G}-\mathbf {F})\mathbf {x}\)  represents the influence of the residual ISI that is not considered by the matrix \(\mathbf {F}\) .
Define the term \(\wp \triangleq \text{Re}\big {\lbrace } \mathbf {e}^\text{H}{\eta } + \mathbf {e}^\text{H}(\mathbf {G}-\mathbf {F})\mathbf {x}\big {\rbrace }\) .
In particular, the term \(\wp \)  can be modeled as a Gaussian variable according to the central limit theorem and it can be shown that \(\mathbb {E}[\wp ] = 0\)  and the variance \(\mathbb {E}[\wp \wp ^{\rm {H}}] = N_0 E_b d^2(\mathbf {e})+2E_b\sigma ^2_{\text{R},\mathbf {e}}\) , where \(\sigma ^2_{\text{R},\mathbf {e}}\)  refers to the variance of the term \(\mathbf {e}^\text{H}(\mathbf {G}-\mathbf {F})\mathbf {x}\)  induced by the residual ISI taps for a given error sequence \(\mathbf {e}\) . In specific, we have \(\sigma ^2_{\text{R},\mathbf {e}} = \frac{1}{2E_b} \mathbb {E}[\mathbf {x}^\text{H}(\mathbf {G}-\mathbf {F})\mathbf {e}\mathbf {e}^\text{H}(\mathbf {G}-\mathbf {F})^\text{H}\mathbf {x}]\)  [1]}.
Based on the derivations in [1]}, the error probability for an ML detection if finite taps of ISI responses are considered by the detector is approximated by the following lemma:

(Error probability for \(\mathbf {e}\)  with finite ISI responses) :
Given the transmitted sequence \(\mathbf {x}\)  and assume that the ML detector considers finite ISI responses with a length of \(L_E\) , the probability of the ML detector declares the sequence \(\mathbf {x}^{\prime } = \mathbf {x} + \mathbf {e}\)  as the detection output is given by
\({P_{\mathbf {e}}^{^{\prime }}} \approx Q\Bigg {(}\sqrt{\frac{{{E_b}d_{{\rm {ope}}}^2(\mathbf {e})}}{{{N_0}}}\frac{{d_{{\rm {ope}}}^2(\mathbf {e})}}{{{d^2}(\mathbf {e}) + \frac{{2\sigma _{{\text{R},\mathbf {e}}}^2}}{{{N_0}}}}}} \Bigg {)}\) .
